A people carrier-type vehicle mounted the kerb at the pedestrian crossing on Tawney Street and Wide Bargate on 10 October, officers say.
The child was clipped and suffered minor injuries.
The collision took place between 12:15 and 12:40pm, and involved a dark blue vehicle, believed to be a Vauxhall seven-seater.
The vehicle turned right from Wide Bargate onto Tawney Street and mounted a kerb where pedestrians were waiting to cross.
A spokesman for Lincolnshire Police said: “Anyone who saw anything, especially if you were driving in the area and/or have dashcam footage, please come forward and let us know on one of the following ways; by ringing 101 quoting incident 233 of 10 October; by emailing PC Thornton on firstname.lastname@example.org; or by calling Crimestoppers, anonymously, on 0800 555111.”